Optical levitation of a non-spherical particle in a ...€¦ · Optical levitation of a...

Optical levitation of a non-spherical particle in a loosely focused Gaussian beam

Cheong Bong Chang, Wei-Xi Huang, Kyung Heon Lee, and Hyung Jin Sung* Department of Mechanical Engineering, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 305-701, South Korea

*[email protected]

Abstract: The optical force on a non-spherical particle subjected to a loosely focused laser beam was calculated using the dynamic ray tracing method. Ellipsoidal particles with different aspect ratios, inclination angles, and positions were modeled, and the effects of these parameters on the optical force were examined. The vertical component of the optical force parallel to the laser beam axis decreased as the aspect ratio decreased, whereas the ellipsoid with a small aspect ratio and a large inclination angle experienced a large vertical optical force. The ellipsoids were pulled toward or repelled away from the laser beam axis, depending on the inclination angle, and they experienced a torque near the focal point. The behavior of the ellipsoids in a viscous fluid was examined by analyzing a dynamic simulation based on the penalty immersed boundary method. As the ellipsoids levitated along the direction of the laser beam propagation, they moved horizontally with rotation. Except for the ellipsoid with a small aspect ratio and a zero inclination angle near the focal point, the ellipsoids rotated until the major axis aligned with the laser beam axis.

©2012 Optical Society of America

OCIS codes: (350.4855) Optical tweezers or optical manipulation; (000.4430) Numerical approximation and analysis; (140.7010) Laser Trapping.

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#172979 - $15.00 USD Received 20 Jul 2012; revised 21 Sep 2012; accepted 25 Sep 2012; published 5 Oct 2012(C) 2012 OSA 8 October 2012 / Vol. 20, No. 21 / OPTICS EXPRESS 24068

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1. Introduction

Optical forces induced by laser beams have been used to manipulate particles or biological cells under a picoNewton-scale force exerted by a laser beam on a micro-sized object capable of moving or trapping the object [1–3]. An optical force acts on the surfaces of transparent objects due to the momentum change of the light induced by the laser beam reflection and refraction, which depends on the shape, size, elasticity, refractive index, inclination angle, and position of the object. The development or improvement of optical particle manipulators and separators that rely on these forces requires an understanding of the behaviors of an object under certain conditions.

A variety of experimental methods have been developed for optical manipulation applications. Among these methods, optical levitation and optical trapping are first reported by Askin and coauthors [4,5]. An individual particle or cell may be manipulated by optical trapping using one or two laser beams [5,6]. Multiple particles or cells may be separated using optical chromatography (OC) and cross-type optical particle separator (COPS) [7–10]. Theoretical studies have been conducted in conjunction with these experiments. Three methods were generally used to calculate the optical force. The optical force exerted on an object much smaller than the laser beam wavelength was calculated by the Rayleigh dipole approximation [11], while the optical force exerted on an object much larger than the laser beam wavelength was calculated by the ray optics method [6,12–20]. For arbitrary size including the intermediate size, the generalized Lorenz-Mie theory was employed to calculate the optical force [21–23]. Among those methods, the ray optics method is relatively simple


and most biological cells, i.e., red blood cells and leukocytes, fall within the size range described by the ray optics method.

The ray optics method decomposes an incoming light ray into individual rays with direction and momentum. As a ray passes a transparent object, part of the ray is reflected and the remainder is refracted on the object surface due to the differences between the refractive indices of the media on either side of the surface. The direction and momentum of the incident ray change during reflection and refraction. Hence, each ray must be traced. Spherical particles are regular in shape, and the geometric relation may be used to obtain the momentum transfer [6,12–15]. This approach is referenced in the present study as the traditional ray optics (TRO) method. Non-spherical particles or continuously evolving elastic materials cannot be described using geometric relations. Gauthier derived the optical forces on a micromotor, a ring-shaped object, a cylindrical object, and a red blood cell by tracing the individual ray vectors [16–18]. This method is applicable to well-characterized shapes. The optical force on an object with a more complex geometry may be obtained using the dynamic ray tracing (DRT) method developed by Sraj et al. [19]. This method involves tracing each ray using the ray-triangle intersection algorithm [24]. Sraj et al. [19] conducted simulations of an elastic capsule subjected to a linear diode bar and described the deformation of the elastic capsule within the fluid–membrane coupled system.

Under a laser beam, non-spherical particles experience torque in addition to the optical force. An optically-induced torque is generated as a result of imbalances in the optical force profile, which depends on the shape of the particle or the laser beam intensity distribution. The rotation induced by the torque is an important factor for controlling an object’s motion because the optical force varies with the incident angle during rotation. Optically trapped objects were considered in several experimental and numerical studies, which revealed that the major axis of an elliptical particle or a biological cell aligns along the direction of the laser beam propagation under a rotational torque [18,20,25–28]. Moreover, it was shown that the trapping force varied with the shape and orientation of the elliptic particle [20,27–30]. Grover et al. [18] examined the behavior of trapped red blood cells, whereas previous studies have mainly focused on the equilibrium state in a tightly focused laser beam. A loosely focused laser beam, such as are used for optical particle separation, i.e. OC and COPS, illuminates a cross-sectional area wider than the object size; hence, until an equilibrium state is reached, the object undergoes horizontal movement with rotation in a wider region because the optical force and torque change continuously during rotation. The optical force distribution under a loosely focused laser beam was statically analyzed and sufficiently characterized for a spherical particle [13,14]. The sphere was simply pulled to the laser beam axis during levitation. The motion could be controlled simply by tuning the laser beam intensity and the minimum beam waist radius. However, non-spherical particle behavior depends on the object shape, inclination angle, and position, even with fixed laser beam properties, and these parameters can significantly affect the performance of the particle or cell separator. The static analysis results can predict the motion of an object although analysis cannot exactly resolve the dynamic state during the motion. Dynamic analysis, in combination with static analysis, can reveal the continuous behavior of a non-spherical particle as a function of time, and such characterizations are necessary for understanding the properties of optical levitation.

In the present study, the optical force on a non-spherical particle subject to a loosely focused Gaussian distributed laser beam was calculated using the DRT method. The optical forces and the induced torque on ellipsoids with different aspect ratios were obtained. The effects of the inclination angle and the position from the focal point were considered. The movement of the non-spherical particle in a fluid-particle coupled system was simulated using the penalty immersed boundary method (pIBM) [31]. The fluid flow and the particle motion, including the optical force term, were separately solved, and the terms were coupled by the pIBM. The effects on the behavior of the particles by the inclination angle and the position from the focal point were studied for a variety of ellipsoidal shapes. The behaviors of the


ellipsoids during levitation under the optical force were analyzed based on numerical simulations.

2. Problem formulation and numerical method

2.1. The optical force

The optical force is expressed as the change in the momentum per unit time, i.e.,

.o t

Δ=Δ

pF (1)

In the above equation, Δp is the momentum transfer, which can be expressed as

( )mn c tΔ = ⋅ Δ Δp Q I A where nm denotes the refractive index of the medium, c the speed of

light in a vacuum, Q the dimensionless momentum transfer, and I is the Gaussian beam intensity. In the present study, we assumed that the loosely focused laser beam propagates along the y-axis. The Gaussian beam intensity is functions of y and the radial distance from the focal point,

2

2 2

22exp ,

( ) ( )fdP

Iy yπω ω

= −

(2)

where P is the laser beam power, df the radial distance from the focal point, and ω the waist radius of the laser beam along the y-axis. Moreover, the waist radius of the laser beam may be expressed as

1/22

00 2

0

( ) 1 ,y

yλω ωπω

= +

(3)

where 0λ is the wavelength of the laser beam and 0ω is the minimum waist radius of the laser beam.

The momentum transfer Δp was obtained by calculating the dimensionless momentum transfer Q according to Q = (incident ray)–(reflected ray)–(refracted ray) on the surface intersecting the incident ray. For a spherical transparent object, Q can be obtained simply by using the TRO method due to the simple and regular object geometry [6,14]; however, for non-spherical particles, no simple geometric relation is available for calculating Q. As shown in Fig. 1, the polar angle ( 1φ ) on the lower surface differs from the incident angle (α) of the ray. Moreover, the inclined ellipsoid is not axisymmetric for all axes. Hence, the incident angle on the lower surface must be obtained at each point, and the intersection points between the upper surface and the refracted ray on the lower surface must be determined. To solve this problem, the ray-triangle intersection method was adopted, in which the object surface was discretized into triangular elements, and the element containing the intersection point was identified. The numerical details of the ray-triangle intersection method are described in previous studies [19,24].


Fig. 1. Paths of a ray from the Gaussian laser beam passing through a non-spherical object.

The present study assumed that the initial incident ray from the laser beam on the lower surface hit the barycenter of each triangular element of the object surface. The incident angle

on the surface was obtained according to ( )1cos i iα −= ⋅R N R N , where Ri denotes the

unit directional vector of the incident ray and N denotes the unit normal vector of the surface on the intersection point. It was assumed that all the initial incident rays were propagated along the positive y-axis because the direction of the initial incident rays on the particle surface is only slightly changed in a loosely focused beam. Hence, the unit direction vector of the initial incident ray has only the y-component. The obtained incident angle and the unit normal vector on the intersection points were used to express the unit direction vectors of the refracted and reflected rays as

(2cos ) ,r i α= +R R N (4)

( cos cos ) ,i it i

t t

n n

n nβ α= + − +R R N (5)

where ni and nt are the refractive indices of medium passing the incident ray and the transmitted ray, respectively. β is the refracted angle obtained by Snell’s law, i.e.,

sin sinm pn nα β= , where nm and np are the refractive indices of medium outside and inside

the object, respectively. The dimensionless momentum transfer on the lower surface may be calculated by

( ) (1 ( )) ,l i r tR n Rα α= − − −Q R R R (6)

where the subscript l denotes the lower surface, n the ratio of the refractive indices, i.e. n = np/nm, and R and T are, respectively, the reflectance and transmittance, which are obtained from

( ) ( ) ( )2 2

2 2

1 sin ( ) tan ( ), 1 .

2 sin ( ) tan ( )R T R

α β α βα α αα β α β

− −= + = − + + (7)

During the ray tracing steps, only the refracted ray is traced because the reflectance induced by the small difference between the refractive indices is small [6,13,15]. The intersection point between the upper surface and the refracted ray on the lower surface may be found using the ray-triangle intersection method after knowing the direction vector of the refracted ray and the intersection point with the initial incident ray on the lower surface. The dimensionless momentum transfer Q on the upper surface may be obtained as


(1 ( )) ( ) (1 ( )) ,ti iu i r t

m m m

nn nR R R

n n nα α α

′ ′ ′ ′ ′= − − − −

Q R R R (8)

where the subscript u denotes the upper surface, and α and α′ are the incident angles on the lower and upper surfaces, respectively, as shown in Fig. 1. i′R , r′R , and t′R are, respectively, the unit directional vectors of the incident, reflected, and refracted rays at the upper intersection point obtained from Eqs. (4) and (5). The optical force per unit volume on each element may be calculated by

,g,imo,i i

i

Pn

c A=

ΔF Q (9)

where the subscript i denotes the index of the element and Pg denotes the incident Gaussian-

distributed laser beam power. In computation, Pg is expressed as ( )g,i i e,i iP A= ⋅ ΔI N , for the

ith element on the lower surface of the capsule at which the laser beam is initially incident. On the upper surface, Pg is equal to the value of the initially incident ray on the lower surface, which is refracted on the lower surface and intersects with the current element.

2.2. Fluid-particle interactions

Simulations of the fluid-particle coupled system were conducted by defining the fluid flow on the fixed Eulerian coordinates and the particle motion on the moving Lagrangian coordinates. Fluid flow is governed by the Navier-Stokes (N-S) equations and the continuity equation, i.e.,

20 ,p

tρ μ∂ + ⋅∇ = −∇ + ∇ + ∂

u u u u f (10)

0,∇ ⋅ =u (11)

where 0ρ denotes the fluid density, u the fluid velocity, p the pressure, μ the dynamic viscosity, and f the Eulerian momentum forcing that enforces no-slip conditions along the immersed boundary.

The particle motion is governed by the linear and angular momentum equations, i.e.

( ) 1 2d d ,S

cP o

dV s s

dtρ

Ω

= −U F F (12)

( ) 1 2d d ,S

cP o

dI s s

dtρ

Ω

= × − r F Fω (13)

where (s1, s2) denotes the moving curvilinear coordinates on the particle surface, ρ the density difference between the particle and the fluid, Vp the volume of the particle, Ip the moment of inertia, Uc the center velocity of the particle, ωc the center angular velocity of the particle, r the distance from the particle center to the nodal position, Fo the optical force, and F the Lagrangian momentum forcing. The nodal velocity of the particle is obtained by

.c c= + ×U U rω (14)

For computation, the governing equations are non-dimensionalized using 0ρ as the

characteristic density, Uc as the characteristic velocity, and 0ω as the characteristic length. The non-dimensional form of the N-S equations is written as


*

* * * 2 * **

1,p

Ret

∂ + ⋅∇ = −∇ + ∇ +∂u u u u f (15)

with 0 0cRe Uρ ω μ= , whereas the non-dimensional forms of the continuity equation and the particle motion equations are expressed in the forms given in Eqs. (11), (12), and (13). The star superscript means the dimensionless quantities.

Particle motion is coupled with the fluid flow using the pIBM [31]. The pIBM defines two types of immersed boundaries represented by the massive and massless material points. The twin boundaries are linked by stiff springs with damping so that they behave as a single massive structure. The restoring force is calculated on the Lagrangian immersed boundary points as

( ) ( ) ( )1 2, , ,ib ibs s t tκ= − − + Δ − F X X U U (16)

where κ is a large coefficient, Δt is the computational time step, X and Xib are, respectively, the massive and massless material points, and U and Uib are the corresponding velocities. U is directly obtained from Eq. (14), whereas Uib is calculated from

( ) ( ) ( )( )1 2 1 2, , , , , d ,f

ib s s t t s s tδΩ

= −U u x X x x (17)

where δ() denotes the smoothed approximation of the Dirac delta function. Here, we use the four-point smoothed delta function [32]. X and Xib are, respectively, obtained by

0 0

0 0, d ,

t t

ib ib ibdt t= + = + X X U X X U (18)

The Lagrangian forcing obtained from Eq. (16) is spread over the Eulerian grids near the immersed boundary points using the smoothed delta function,

( ) ( ) ( )( )1 2 1 2 1 2, , , , , d d ,S

t s s t s s t s sδΩ

= −f x F x X (19)

which is added to the N-S equations. In the framework of the pIBM, the governing equations for the fluid flow and the particle

motion are separately solved using different numerical methods. In the present study, the N-S equations and the continuity equation are solved using the fractional step method on a staggered Cartesian grid system [33], whereas the linear and angular momentum equations of the particle are solved using the Runge-Kutta procedure [34], with the particle surface discretized into triangular elements using the subdivision surface method [35]. The pIBM implementation is described in detail in the computational procedure of Huang et al. [31].

3. Results and discussions

3.1. Static analysis of the optical force

The optical force on a non-spherical rigid particle was calculated using the DRT method. The effects of the particle shape on the optical force were examined by considering oblate spheroidal particles with aspect ratios of b/a = 1.0, 0.9, 0.5, and 0.3, where a and b denote respectively the major and minor semi-axis as shown in Fig. 2(a). Figure 2(b) shows that the inclination angle θ was defined as the angle from the positive x-axis to the major semi-axis of the ellipsoid. In the spherical case with b/a = 1.0, a was set to be 3.2μm. Because the same surface area was assumed for all particles, a was respectively set to be 3.31μm, 3.85μm, and 4.17μm for the oblate spheroid with aspect ratios of b/a = 0.9, 0.5, and 0.3. Hence, the

equivalent radius defined by 4eqr A π= was 3.2μm. Considering the laser beam with

0 16 mω = μ , the ratio of the equivalent radius of particles to the minimum waist radius of the


laser beam was 0.2, which also means the normalized radius of the particle. Particles with np = 1.40 immersed in the fluid with nm = 1.33 were considered. For the computation, particles were discretized into 8192 triangular elements using the subdivision surface method.

Fig. 2. (a) The oblate spheroid with major and minor semi-axes of a and b, and (b) the inclination angle defined on the x-y plane.

φ1

Qy

0 20 40 60 800

0.1

0.2

0.3(a)

b/a = 0.9

b/a = 0.5

b/a = 0.3

Sphere (DRT)

Sphere (TRO)

φ1

φ 2

0 20 40 60 800

20

40

60

80

(b)

nm = 1.33np = 1.40

Fig. 3. (a) The y-component of the dimensionless momentum transfer, and (b) the matched polar angles on the upper and lower surfaces.

Prior comparing the optical forces, the y-component of the dimensionless momentum transfer (Qy = Qu,y + Ql,y) at the focal point was obtained as a function of the polar angle ( 1φ ) on the lower surface, as shown in Fig. 3(a). One can see that Qy reached its maximum at a high polar angle. As the aspect ratio of the ellipsoid decreased, the maximum value of Qy occurred at higher polar angles and became larger than the value for a sphere because the larger incident angle induced larger momentum changes. The polar angle on the lower surface of the ellipsoid was not coincident with the incident angle. The high incident angle was located at the high polar angle, but the region encompassing the high incident angle decreased as the aspect ratio decreased, thereby decreasing Qy as a whole. The present method was validated by comparing the results with those obtained using the TRO method for the spherical case. Good agreement was observed in Fig. 3(a).

Figure 3(b) shows the polar angle ( 2φ ) at the intersection points on the upper surface corresponding to the first intersection point on the lower surface during ray tracing, as a function of the polar angle ( 1φ ) on the lower surface. The polar angles on the lower and upper surfaces are indicated in Fig. 1. The initial incident rays were refracted and reflected on the lower surface. Then, the refracted rays were intersected on the upper surface. In the spherical case, the upper surface points within the range of 2φ = 54-59° were illuminated by the

refracted rays at two different lower surface points with high polar angle ( 1φ ), whereas the


region containing the higher polar angles ( 2φ ) on the upper surface was not illuminated by the refracted rays from the lower surface. The trend in the spherical case resembled the trend reported by Bareil et al. [15], and the results by both the DRT and TRO methods agreed well. As the aspect ratio of the ellipsoid decreased, the polar angle 2φ illuminated by the refracted

rays increased and gradually becomes monotonic as a function of 1φ . For b/a = 0.3, 2φ is was

almost equivalent to 1φ , as shown in Fig. 3(b).

Fig. 4. Distribution of the optical force on rigid particles of different shapes: (a) sphere; (b) ellipsoid with b/a = 0.9; (c) ellipsoid with b/a = 0.5; (d) ellipsoid with b/a = 0.3.

The optical force distribution and the total y-component optical force at the focal point were calculated for a variety of shapes of the particles, as shown in Fig. 4. We assumed a laser beam characterized by P = 0.5 W and 0λ = 1064 nm. The magnitude of the optical force at each node was obtained by

2 2 2, , , , , , , .o i o x i o y i o z i iF F F F A= + + Δ (20)

The total x- and y-components of the optical force were, respectively, calculated according to

, , , , , ,1 1

, .NE NE

o x o x i i o y o y i ii i

F F A F F A= =

= Δ = Δ (21)

The optical forces were non-dimensionalized by nmP/c, which described the dimensionless momentum transfer. The maximum optical force on the sphere and the ellipsoid with b/a = 0.9 was observed near the middle latitude of the upper surface. The optical force was almost negligible from the equator of the particle to the latitude of the maximum value. As expected based on Fig. 3(b), the maximal optical force was induced by the multiple rays illuminated from the lower surface to the upper surface. As the aspect ratio of the ellipsoid decreased, the maximum ring-shaped distribution of the optical force disappeared and the non-illuminated region was reduced because the rays between the lower and upper surfaces could be matched through a one-to-one correspondence, as shown in Fig. 3(b). The total y-component of the optical force decreased as the aspect ratio decreased due to the decrease in the dimensionless momentum transfer, indicating that the near-spherical particles were levitated to a greater extent by the optical force.


y/ω0

F

o,y/(

n mP

/c)

0 100 200 300 400 5000

0.001

0.002

Sphere(a)

x = 0

y/ω0

F

o,y/(

n mP

/c)

0 100 200 300 400 5000

0.001

0.002

0π/6π/3π/2

-π/6-π/3

(c)x = 0 b/a = 0.9

x/ω0

F

o,x/(

n mP

/c)

-3 -2 -1 0 1 2 3

-0.001

0

0.001

(b)y = 0 Sphere

x/ω0

F

o,x/(

n mP

/c)

-3 -2 -1 0 1 2 3

-0.001

0

0.001

(d)y = 0 b/a = 0.9

Fig. 5. The optical forces on the particles with different inclination angles as a function of the x- and y-directional positions: (a) the y-component optical force on a sphere; (b) the x-component optical force on a sphere; (c) the y-component optical force on an ellipsoid with b/a = 0.9; (d) the x- component optical force on an ellipsoid with b/a = 0.9.

The total x- and y-components of the optical force on particles with different aspect ratios, along with the inclination angles, were calculated as functions of the position, as shown in Figs. 5 and 6. As shown in the spherical case, in Figs. 5(a) and 5(b), the y-component of the optical force reached its maximum at the focal point and pushed the particle along the direction of the laser beam propagation, whereas the x-component optical force was zero at the focal point and pulled the particle toward the laser beam axis. For an ellipsoid with b/a = 0.9, the y-component optical force followed the same trend observed in the spherical case and varied only slightly with the inclination angle; however, the x-component of the optical force was generally non-zero at the focal point, although the inclined ellipsoid was pulled toward the laser beam axis. The ellipsoid with counterclockwise rotation experienced a balanced position along the x-direction on the left-hand side of the laser beam axis, and vise versa. Here, the balanced position means the position with zero net force.

The optical force distributions on ellipsoids with small aspect ratios were also considered. As shown in Figs. 6(a) and 6(c), the overall trend in the y-component optical force distributions resembled the trend in the spherical case, but the magnitudes differed. A slight inclination of an ellipsoid with b/a = 0.5 yielded a y-component the optical force with a magnitude comparable to that of the non-inclined ellipsoid but less than that of the sphere. The ellipsoid with an inclination angle of π/2 experienced similar values of the y-component optical force; however, the optical force on the significantly inclined ellipsoid increased to a great extent because the region characterized by large incident angles was widened. The ellipsoid with b/a = 0.3 behaved similarly to the ellipsoid with b/a = 0.5, but the magnitude of the y-component optical force was reduced. The x-component optical force distributions on ellipsoids with b/a = 0.5 and 0.3 differed significantly from the distribution on an ellipsoid with b/a = 0.9. The significantly inclined ellipsoids with b/a = 0.5 and 0.3 did not experience a balanced position along the x-axis. The ellipsoid with counterclockwise rotation always experienced a negative x-component optical force, whereas the ellipsoid with clockwise rotation always experienced a positive x-component optical force. The results show that the object with a refractive index larger than that of the outer medium was not always pulled toward the laser beam axis by the rotation of the ellipsoid. As the aspect ratio of the ellipsoid decreased, the optical force on the slightly inclined ellipsoid decreased. In contrast, the optical force on the significantly inclined ellipsoid increased.


y/ω0

F

o,y/(

n mP

/c)

0 100 200 300 400 5000

0.001

0.002(a)x = 0 b/a = 0.5

0π/6π/3π/2

-π/6-π/3

y/ω0

F

o,y/(

n mP

/c)

0 100 200 300 400 5000

0.001

0.002(c)x = 0 b/a = 0.3

x/ω0

F

o,x/(

n mP

/c)

-3 -2 -1 0 1 2 3

-0.004

-0.002

0

0.002

0.004(b)

y = 0 b/a = 0.5

x/ω0

F

o,x/(

n mP

/c)

-3 -2 -1 0 1 2 3

-0.004

-0.002

0

0.002

0.004(d)

y = 0 b/a = 0.3

Fig. 6. The optical forces on particles with different inclination angles, as a function of the x- and y-directional positions: (a) the y-component optical force on an ellipsoid with b/a = 0.5; (b) the x-component optical force on an ellipsoid with b/a = 0.5; (c) the y-component optical force on an ellipsoid with b/a = 0.3; (d) the x-component optical force on an ellipsoid with b/a = 0.3.

The total x-component of the optical force was zero at the focal point only for the sphere and the ellipsoid with an inclination angle of zero or π/2. The optical forces on the ellipsoids with different inclination angles were generally not balanced. Hence, the optically induced z-directional torque needed to be accounted for, which was obtained by

( ), , , , , ,1

,NE

z x i o y i y i o x i ii

r F r F Aτ=

= − Δ (22)

where rx and ry are respectively the x- and y- component distances between the element center and the particle center. The optically induced torque was non-dimensionalized by nmPreq/c. Figure 7 shows that the torque was zero for ellipsoids with an inclination angle of zero or π/2, located at the focal point; however, the ellipsoids experienced torque near the laser beam axis under the Gaussian distribution of the laser beam, and the magnitude of the torque depended on the aspect ratio. As shown in Fig. 7(a), the torque on the sphere was so small that the rotation was negligible. The torque on the ellipsoids changed significantly with the inclination angle because the optical force and the moment arm changed under the rotation of the ellipsoid. For b/a = 0.9, the ellipsoid with counterclockwise rotation experienced a positive torque at all positions, whereas the ellipsoid with a clockwise rotation experienced a negative torque at all positions. As the aspect ratio decreased, the torque on a slightly inclined ellipsoid approached zero near the laser beam axis because the moment arm of the x-component optical force was small; however, the largely inclined ellipsoid experienced a large torque due to the large moment arm and the large optical force along the x-axis. In sum, a near-spherical particle at the focal point experienced the large y-component optical force, whereas the significantly inclined ellipsoid with a small aspect ratio experienced the large x- and y- component optical force and torque, and was repelled from the laser beam axis during rotation.


x/ω0

τ z/(n m

Pr eq

/c)

-3 -2 -1 0 1 2 3

-1.5E-06

0

1.5E-06(a)

y = 0 Sphere

x/ω0

τ z/(n m

Pr eq

/c)

-3 -2 -1 0 1 2 3-0.0025

0

0.0025

0.005(c)y = 0 b/a = 0.5

x/ω0

τ z/(n m

Pr eq

/c)

-3 -2 -1 0 1 2 3-0.0005

0

0.0005(b)

y = 0 b/a = 0.9

0π/12π/4

5 π/12π/2

-π/4

x/ω0

τ z/(n m

Pr eq

/c)

-3 -2 -1 0 1 2 3-0.0025

0

0.0025

0.005(d)y = 0 b/a = 0.3

Fig. 7. The z-component torque on particles with different inclination angles, as a function of the x-directional position: (a) sphere; (b) ellipsoid with b/a = 0.9; (c) ellipsoid with b/a = 0.5; (d) ellipsoid with b/a = 0.3.

3.2. Dynamic analysis of the optical force

The static analysis of the optical force and induced torque described above revealed that a sphere positioned near the laser beam was always pushed along the direction of the laser beam propagation and was always pulled toward the laser beam axis; however, non-spherical particles with different aspect ratios displayed different optical force and torque distributions according to the inclination angle. Moreover, the inclination angle varied continuously during rotation. Hence, for the non-spherical particles, a dynamic analysis was necessary. In the present study, the motions of ellipsoids with a dimensionless density difference *ρ = 0.1 in an initially stationary fluid with dimensional parameter of ρ0 = 1000 kg/m3 and μ = 0.001 kg/m·s were simulated. A fluid domain of * * *20 40 20eq eq eqr r r× × was used along the x-, y-, and

z-directions and was discretized by a 128 × 256 × 128 grid. Uniform grids were used along all of the directions. The origin of the coordinate in the y-direction was positioned a distance of 10 *

eqr from the bottom, whereas the origin of the coordinates in x- and z- directions was

positioned at the center of the domain. The laser beam was focused on the origin. The Reynolds number, based on the particle vertical velocity Vc in the final steady state, was about Re = 0.0005 in our simulations. For computation of fluid-particle coupled system, all the variables were non-dimensionalized by the characteristic scales mentioned in section 2.2. However, for comparison with the result in section 3.1, the obtained optical forces and torque were again non-dimenaionalized by nmP/c and nmPreq/c, respectively. The computational time t was non-dimensionalized by ω0/Vc. In all simulations, the computational time step was *tΔ

= 0.00002 and the free constant in Eq. (16), 77.8 10κ = × , was adopted. The simulation results for ellipsoids with different aspect ratios and inclination angles at

different positions are shown in Figs. 8–13. As shown in Fig. 8, ellipsoids with an inclination angle of π/4 at the focal point were considered because the x-component optical force on ellipsoids with an inclination angle of zero or π/2 was balanced at the focal point. Figures 8(a), 8(b), and 8(c) show the instantaneous shapes of the ellipsoids for b/a = 0.9, 0.5, and 0.3. The solid line and dotted line, respectively, indicate the positions of the laser beam axis and the trajectory of the ellipsoid center. As shown in Figs. 5-7, the ellipsoids with counterclockwise rotation experienced the negative x-component optical force and the


positive torque. Hence, the ellipsoids initially shifted toward the left-hand side of the laser beam axis with counterclockwise rotation. As the inclination angle approached π/2, the major axes aligned with the laser beam axis, and the ellipsoids shifted toward the laser beam axis without rotation. During the rotation and horizontal movements, the ellipsoids were always pushed along the direction of the laser beam propagation under a positive y-component optical force. The time histories of the inclination angles, the total x-component optical force, and the optically-induced torque are plotted in Figs. 8(d), 8(e), and 8(f), respectively. As shown in Fig. 8(d), all ellipsoids rotated until the inclination angle reached π/2. The ellipsoid with b/a = 0.9 rotated gradually, whereas the ellipsoids with b/a = 0.5 and 0.3 rotated rapidly. Variations in the optical force and torque (Figs. 8(e) and 8(f)) were consistent with the static analysis of the continuous changes in the position and inclination angle. As shown in Fig. 8(f), the ellipsoid with a small aspect ratio experienced a large torque during rotation because the torque increased under the higher x-component optical force and moment arm. As the ellipsoids aligned with the laser beam axis, the torque decreased. The ellipsoids that deviated from an inclination angle of π/2 were restored to the π/2 inclination angle under the damping effects of the fluid viscosity and the optically-induced torque, as indicated in the static analysis. These results were similar to the results for an ellipsoidal particle under optical trapping, reported previously [18,20,25–28]. Here, dynamic analysis in the presence of a loosely focused laser beam induced horizontal motion during rotation.

Fig. 8. Ellipsoids with an inclination angle of π/4, released at the focal point: the instantaneous shapes and trajectories of the ellipsoids with (a) b/a = 0.9, (b) b/a = 0.5 and (c) b/a = 0.3; the time histories of (d) the inclination angle, (e) the total x-component optical force, and (f) the optically-induced torque.

Ellipsoids released at some horizontal distance from the laser beam axis were also considered. The horizontal distance was normalized by the minimum waist radius of the laser beam ω0. The distances of 0.5 and 1.0 were considered, which respectively denote ω0/2 and ω0. Figure 9 shows the results for ellipsoids with an inclination angle of π/4, released at a horizontal distance of 0.5 from the focal point. As shown in Figs. 9(a) and 9(e), the ellipsoid with b/a = 0.9 was pulled toward the laser beam axis because the ellipsoid experienced a positive x-component optical force near the laser beam axis. The ellipsoid rotated gradually until the major axis aligned with the laser beam axis, as shown in Figs. 9(a) and 9(d). The ellipsoids with b/a = 0.5 and 0.3 initially shifted toward the left with a counterclockwise rotation. As the ellipsoids aligned with the laser beam axis, they were pulled back toward the laser beam axis without rotation then were simply pushed along the laser beam propagation


direction. As shown in Figs. 9(e) and 9(f), the trends in the optical force and induced torque were the same as the trends observed at the focal point.

Fig. 9. Ellipsoids with inclination angles of π/4, released at a horizontal distance of 0.5 from the focal point: the instantaneous shapes and trajectories of the ellipsoids with (a) b/a = 0.9, (b) b/a = 0.5, and (c) b/a = 0.3; the time histories of (d) the inclination angle, (e) the total x-component optical force, and (f) the optically-induced torque.

Fig. 10. Ellipsoids with an inclination angle of -π/4, released at a horizontal distance of 0.5 from the focal point: the instantaneous shapes and trajectories of the ellipsoids with (a) b/a = 0.9, (b) b/a = 0.5, and (c) b/a = 0.3; the time histories of (d) the inclination angle, (e) the total x-component optical force, and (f) the optically-induced torque.

The ellipsoid with a negative inclination angle at the focal point behaved symmetrically with respect to the y-axis, as compared with the ellipsoid with a positive inclination angle, because the optical force and torque on an ellipsoid with a positive or negative inclination angle were symmetric at the focal point, as shown in Figs. 5–7; however, the ellipsoid with a negative inclination angle that initially deviated from the focal point showed different


behaviors. As shown in Figs. 5–7, the ellipsoids with b/a = 0.9 and 0.5, and with negative inclination angles, generally experienced a positive x-component optical force and a negative torque. Hence, the ellipsoids with an inclination angle of −π/4 shifted toward the laser beam axis under rotation, as shown in Fig. 10(a) and 10(b). The ellipsoids were then aligned with and moved along the laser beam axis. The ellipsoid with b/a = 0.3 continuously experienced a large positive x-component optical force and was pulled from the left toward the right-hand side of the laser beam axis. This ellipsoid rotated in the counterclockwise direction on the left-hand side and in the clockwise direction on the right-hand side. Finally, the ellipsoid with b/a = 0.3 was pulled back toward the laser beam axis at the right-hand side with an inclination angle of π/2, as shown in Fig. 10(c). Figure 10(d) shows that the inclination angles were converged to π/2. As compared with the ellipsoids with the initial inclination angle of π/4, the reversed optical force and torques were shown in Figs. 10(e) and 10(f).

Fig. 11. Ellipsoids with an inclination angle of π/2, released at a horizontal distance of 0.5 from the focal point: the instantaneous shapes and trajectories of the ellipsoids with (a) b/a = 0.9, (b) b/a = 0.5, and (c) b/a = 0.3; the time histories of (d) the inclination angle, (e) the total x-component optical force, and (f) the optically-induced torque.

The behavior of the ellipsoid with an inclination angle of π/2 released at a horizontal distance of 0.5 from the focal point was examined. As shown in Fig. 11, all ellipsoids were simply pulled toward the laser beam axis without rotation and were pushed along the laser beam propagation direction. This behavior resembled that observed in the final states of the initially inclined ellipsoids. The non-inclined ellipsoids at the same positions were considered. Figures 12(a) and 12(b) show that the ellipsoids with b/a = 0.9 and 0.5 were pulled toward the laser beam axis under rotation, similar to the behavior shown in Figs. 8(a) and 8(b), because the ellipsoids rotated significantly near the focal point under the Gaussian-distributed laser beam; however, the ellipsoid with b/a = 0.3 shifted toward the laser beam axis with negligible rotation because the torque on the slightly inclined ellipsoid with b/a = 0.3 approached zero near the laser beam axis, as shown in Fig. 7(d). Hence, the ellipsoid did not experience sufficient torque to overcome the fluid viscous forces and reach an inclination angle of π/2. Because the optical force and the torque on the slightly inclined ellipsoids increased with the aspect ratio, as shown in Figs. 6 and 7, the ellipsoids with b/a = 0.9 and 0.5 were able to rotate near the laser beam axis.


Fig. 12. Ellipsoids with an inclination angle of zero, released at a horizontal distance of 0.5 from the focal point: the instantaneous shapes and trajectories of the ellipsoids with (a) b/a = 0.9, (b) b/a = 0.5, and (c) b/a = 0.3; the time histories of (d) the inclination angle, (e) the total x-component optical force, and (f) the optically-induced torque.

Fig. 13. Ellipsoids with b/a = 0.3 and different inclination angles, released at a horizontal distance of 1.0 from the focal point: the instantaneous shapes and trajectories of the ellipsoid with (a) an inclination angle of zero, (b) an inclination angle of π/4, and (c) an inclination angle of -π/4; the time histories of (d) the inclination angle, (e) the total x-component optical force, and (f) the optically-induced torque.

Ellipsoids with b/a = 0.3 and with different inclination angles and released at a horizontal distance of 1.0 from the laser beam axis were considered. The ellipsoid with an inclination angle of zero approached the laser beam axis initially, then moved far away due to the counterclockwise rotation. Finally, the ellipsoid moved back toward the laser beam axis with an inclination angle of π/2. An inclination angle of π/4 (Fig. 13(b)) resulted in the rapid rotation of the ellipsoid to π/2, followed by a shift toward the laser beam axis without rotation. The behavior of the ellipsoid with an inclination angle of –π/4 (Fig. 13(c)) resembled


that shown in Fig. 10(c). The ellipsoid aligned along the laser beam axis at the right-hand side of the focal point. Figure 13(d) shows that all ellipsoids ultimately displayed an inclination angle of π/2. As shown in Figs. 13(e) and 13(f), the ellipsoid with an inclination angle of π/4 experienced a small optical force and torque relative to a non-inclined ellipsoid because the initially non-inclined ellipsoid moved toward and rotated near the laser beam axis.

Previous studies of trapped red blood cells revealed that a stable equilibrium could be obtained at an inclination angle of π/2 and an unstable equilibrium could be obtained at an inclination angle of zero [18]. The results obtained in the present study were generally the same; however the non-inclined ellipsoid with a small aspect ratio near the focal point maintained a zero inclination angle because the disturbance, i.e., the optically-induced torque, was insufficient to overcome the damping effects of the fluid viscosity at small inclination angles near the focal point. The dynamic analysis showed that the entire process of ellipsoid movement could be resolved. Until the ellipsoids reached a final state, they were levitated with a sideways displacement at a certain inclination angle during rotation, indicating that an object with a higher refractive index than that of the outer medium could be repelled from the laser beam axis; however, the rotation continuously changed the inclination angle. After reaching a final state, the ellipsoids were pulled toward the laser beam axis without rotation during levitation.

4. Conclusions

The optical forces on ellipsoidal particles subjected to a Gaussian-distributed laser beam were calculated using the DRT method. The optical force was induced by interactions between a transparent object and the laser beam; therefore, the effects of the shapes and the inclination angles on the optical force were examined in detail. As the aspect ratio of the ellipsoid decreased, the total y-component optical force decreased because the region illuminated by the rays with a high incident angle decreased in size. A ring-shaped region showing a peak optical force was observed on the upper surface of the spheres or the ellipsoids with a large aspect ratio. This region was absent from ellipsoids with small aspect ratios. The incident angle on the surface varied with the inclination angle of the particle; therefore, ellipsoids with different inclination angles at a variety of positions were considered. Both the near-spherical particles and the significantly inclined ellipsoids experienced large y-component optical forces. Ellipsoids with an inclination angle of zero or π/2 experienced a balance of optical forces along the x-component at the focal point. By contrast, ellipsoids with different inclination angles experienced an x-component optical force and torque and were not always pulled toward the laser beam axis. A dynamic analysis revealed the continuous change in position, the inclination angle, the optical force, and the torque on the ellipsoids. The motion of the ellipsoids in a stationary fluid was simulated using the pIBM to solve the fluid–particle coupled system. Ellipsoids with different initial inclination angles were considered at different initial positions. All ellipsoids gradually moved along the direction of the laser beam propagation. The ellipsoids also rotated until their inclination angle reached π/2, except for the ellipsoid with a small aspect ratio and zero initial inclination angle near the laser beam axis. During rotation, the inclined ellipsoids with small aspect ratios were pulled toward or repelled away from the laser beam axis, depending on the position and inclination angle. After rotation, the ellipsoid simply moved toward the laser beam axis. The dynamic analysis was consistent with the static analysis by changing the parameters continuously. An understanding of the behavior of ellipsoids subjected to a laser beam can be used to improve the efficiency of particle or cell separation applications using optical forces.

Acknowledgment

This study was supported by the Creative Research Initiatives (No. 2012-0000246) program of the National Research Foundation of Korea.


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